Now that populations in advanced societies are hovering near the replacement level, it is important that demographic analysis push beyond the standard linear analysis associated with the school of Lotka. Our preliminary findings with respect to deterministic non-linear marriage and other two-sex models motivate the importance of the first half of the proposed analytical population research--that dealing with first-degree-homogeneous, non-linear, non-monotone relationships that go beyond and complete the models associated in the last two decades with D. Kendall, L. Goodman, and J. Pollard. In particular, the research will elucidate an important phenomenon not hitherto recognized, namely the distinct possibility that a population which is no longer increasing may generate a self-aggravating distortion of the sex ratio. The second half of our researches, which hope to bring to bear on demographic analysis the tools developed in econometrics and mathematical economics of the last third of the century, has to do with stochastic demographic models. In particular, first-degree-homogeneous probability generalizations of the deterministic models lead to important phenomena different from the Galton-Watson processes that are properly given an important role in stochastic population analysis. As a typical finding, we can show that when the mean value of the population is at the replacement level with no tendency toward decay, it nevertheless will follow that with probability going to one the population will ultimately die out (as when the median and every quantile value decays exponentially). This is the phenomenon expected with log-normal and other multiplicative processes. Included in these researches are models that involve interrelations among different classes, sexes, species (as in predator-prey models). It is expected that the combination of the skills of the economist and the physicist will result in several publications beyond findings already secured.